..Konoha's White Fang..
Sarutobi along with his former teammates and now the Village Elders, have known about Akatsuki since long before Jiraya. During the Sandaime's early years of leadership, he and his advisory thought it best for Akatsuki to be wiped out early.. before it could grow into the overwhelmingly powerful group it is today.
Their original plan was to infiltrate a Hokage level ninja into the organization, and relay enough inside information for the Sarutobi's special ANBU to come up with an eradication method. This plan left only 4 ninja to choose from: The Three Sannin of the Leaf, and Konoha's White Fang.
Orochimaru was the least known of the Sannin (at the time) and by far the best choice for the job, being the strongest of the 4 and just naturally a shady m*therfucker. He'd fit in with the Akatsuki almost alarmingly well. Of course, as it would turn out, the ANBU discovered one of his secret experiments when told to search him out and notify him of the mission.. which would eventually lead to... well you know.
With Orochimaru's crazy ass out of the picture, this left only the Toad Sage, Slug Bitch, Konoha's White Fang to choose from for the mission. Well Jiraya and Tsunade being too well known, and generally feared by any wicked ninja, only Konoha's White Fang was left to choose from. So now that they had their man, Sarutobi staged the White Fang's death and sent him on his mission.
All had been going to plan, or so they thought. Konoha hadn't expected Madara to anticipate Hatake's motives so keenly, and turn the man into a double agent. The White Fang had been ordered to recruit his son into the organization, or be killed and have his son recruited anyway.
--------------------------------------------------------------------------
..Kakashi..
Kakashi was recruited shortly after Hatake Sakumo 'died' and joined the Akatsuki. His father found the time to train him when Kakashi would take solo missions and the organization would send one of their minions under 'mind control' to complete the mission for him. (hence his promotion to jonin at age 13)
Kakashi's role in the organization has always been to keep an eye on Konoha, especially the Yondaime, and has been used by Madara to very discreetly prepare Minato to be ready to seal the Kyuubi when it came time for Madara to summon him. The original plan was to steal the Yondaime's corpse the moment he sealed the Bijuu into himself, but their plans were obviously thrown off course when Akatsuki realized Minato had advanced his sealing jutsu to unexpected heights.
--------------------------------------------------------------------------
Unknown to the rest of the organization, Kakashi's Father's plans were also thrown severely off course. He'd originally planned to break the Akatsuki's hold on Kakashi after the Kyuubi had been captured. Upon hearing of Akatsuki's failure to capture the 9 Tailed Fox, The White Fang became enraged.
He attacked the organization from within, killing most of it's members before being slaughtered by Pein and turned into one of his backup bodies. Though he pretty much succeeded at destroying the organization. He left Akatsuki with a member count of 4, well 3 if you don't count Kakashi because he too attacked Pein, wanting revenge for his father's death. Pein had no choice but to seal the child's mind and leave him to grow as a Leaf Shinobi, only to be used again when the time was right. When Akatsuki could rebuild and resurrect their plans to capture the Tailed Beasts.
Soon after, Sarutobi was forced to find yet ANOTHER candidate to infiltrate Akatsuki. Though now, he had other things on his mind such as the possibility of a rebellion of the Uchiha Clan. So... figuring he could kill 2 birds with 1 stone, Sarutobi and the Village Elders ordered Uchiha Itachi to slaughter his clan before infiltrating the Akatsuki. It had to be this way, or Akatsuki would have questioned Itachi's motives.
... O_O ...